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Derivation

Primordial Helium from Big Bang Nucleosynthesis

Statement

In the radiation-dominated early universe the neutron-to-proton ratio is held in equilibrium by the charged-current weak interactions \(n+\nu_e\rightleftharpoons p+e^-\) and \(n+e^+\rightleftharpoons p+\bar\nu_e\). When the weak reaction rate falls below the Hubble expansion rate the ratio freezes out at \(n_n/n_p\simeq \exp(-\Delta m\,c^2/k_BT_f)\approx 1/5\) near \(k_BT_f\approx 0.8\ \mathrm{MeV}\); residual free-neutron \(\beta\)-decay before the deuterium bottleneck opens lowers it to \(\approx 1/7\); and, because essentially every surviving neutron is locked into \({}^4\mathrm{He}\), the primordial helium mass fraction is \(Y=\dfrac{2(n_n/n_p)}{1+(n_n/n_p)}\approx 0.25\).

Why it matters

The value \(Y\approx 0.25\) is one of the oldest quantitative predictions of hot Big Bang cosmology, fixed within the first three minutes and preserved essentially unchanged for \(13.8\) billion years. It is remarkable that a single dimensionless number follows from three separately measured microphysical inputs — the neutron-proton mass splitting, the Fermi weak coupling, and the neutron lifetime — combined with the Friedmann expansion law. Its agreement with the observed helium abundance in metal-poor gas is a pillar of the standard cosmological model.

Because \(Y\) depends on the expansion rate at \(k_BT\sim 1\ \mathrm{MeV}\), which in turn depends on the number of relativistic species \(g_*\), primordial helium is also a precision probe of physics beyond the Standard Model: it bounds the effective number of neutrino families and any extra light degrees of freedom present one second after the Big Bang.

Assumptions
Thermal equilibrium and Maxwell-Boltzmann nucleon statistics.If nucleons were not in kinetic and weak chemical equilibrium above freeze-out, the ratio \(n_n/n_p\) would not be the simple Boltzmann factor \(e^{-\Delta m c^2/k_BT}\) and the entire initial condition for the abundance would be undetermined.
Standard radiation-dominated Friedmann expansion with \(g_*\approx 10.75\).If \(H(T)\) differed — extra relativistic species, modified gravity, or early matter domination — the freeze-out temperature \(T_f\propto (\sqrt{g_*}/G_F^2 M_{\rm Pl})^{1/3}\) would shift and with it \(Y\).
Negligible lepton asymmetry: \(\mu_{\nu_e}\approx 0\).A large electron-neutrino chemical potential would bias the \(\beta\)-equilibrium, multiplying the ratio by \(e^{-\mu_{\nu_e}/k_BT}\) and changing \(Y\); observationally the asymmetry is tiny.
The deuterium bottleneck delays nucleosynthesis to \(k_BT_{\rm nuc}\approx 0.07\ \mathrm{MeV}\).The baryon-to-photon ratio \(\eta\sim 6\times10^{-10}\) is so small that the high-energy photon tail photodissociates \({}^2\mathrm{H}\) until \(k_BT\ll B_D\); dropping this delay would leave far more free neutrons at onset, over-producing helium.
Every neutron that survives to \(T_{\rm nuc}\) ends in \({}^4\mathrm{He}\).\({}^4\mathrm{He}\) is by far the most bound light nucleus, so the reaction network funnels neutrons into it; the residual free-neutron and trace \({}^2\mathrm{H},{}^3\mathrm{He},{}^7\mathrm{Li}\) fractions are \(\lesssim10^{-4}\) and correcting for them shifts \(Y\) only at the percent level.
Derivation
1
\[ \frac{n_n}{n_p}=\frac{g_n\,(m_nk_BT)^{3/2}e^{(\mu_n-m_nc^2)/k_BT}}{g_p\,(m_pk_BT)^{3/2}e^{(\mu_p-m_pc^2)/k_BT}} \]
Non-relativistic equilibrium number densities for both nucleons at common temperature \(T\); \(g_n=g_p=2\). B
2
\[ \frac{n_n}{n_p}=\left(\frac{m_n}{m_p}\right)^{3/2}\exp\!\left(-\frac{(m_n-m_p)c^2}{k_BT}\right)\simeq \exp\!\left(-\frac{\Delta m\,c^2}{k_BT}\right) \]
Weak chemical equilibrium \(n+\nu_e\rightleftharpoons p+e^-\) with \(\mu_e,\mu_{\nu}\approx0\) gives \(\mu_n=\mu_p\); the prefactor \((m_n/m_p)^{3/2}=1.002\) is dropped, leaving \(\Delta m c^2=1.293\ \mathrm{MeV}\). B
3
\[ \Gamma_{\rm weak}\;\approx\; \frac{7\pi}{60}\,(1+3g_A^2)\,G_F^2\,(k_BT)^5\;\sim\;G_F^2(k_BT)^5 \]
Thermally averaged charged-current \(n\leftrightarrow p\) rate; phase space of the relativistic leptons gives the \(T^5\) scaling with \(G_F\) the Fermi constant and \(g_A\) the axial coupling. C
4
\[ H=\sqrt{\frac{8\pi G}{3}\rho}\,,\qquad \rho=\frac{\pi^2}{30}\,g_*\,\frac{(k_BT)^4}{(\hbar c)^3}\ \Rightarrow\ H=\sqrt{\frac{4\pi^3 g_*}{45}}\,\frac{(k_BT)^2}{M_{\rm Pl}c^2}\cdot\frac{c^2}{\hbar} \]
Friedmann equation in the radiation era (assumed prior result) with the relativistic energy density; \(M_{\rm Pl}=\sqrt{\hbar c/G}\). B
5
\[ \Gamma_{\rm weak}(T_f)=H(T_f)\ \Rightarrow\ G_F^2(k_BT_f)^5\sim \sqrt{g_*}\,\frac{(k_BT_f)^2}{M_{\rm Pl}c^2} \]
Freeze-out is defined by the crossover of interaction and expansion rates; below \(T_f\) the weak reactions are too slow to track equilibrium. B
6
\[ k_BT_f\;\sim\;\left(\frac{\sqrt{g_*}}{G_F^2\,M_{\rm Pl}c^2}\right)^{1/3}\approx 0.8\ \mathrm{MeV} \]
Solve the crossover for \(T_f\); note the \(1/3\) power makes \(T_f\) insensitive to the order-one coefficients, robustly \(\sim1\ \mathrm{MeV}\). C
7
\[ \left.\frac{n_n}{n_p}\right|_{f}=\exp\!\left(-\frac{\Delta m c^2}{k_BT_f}\right)=\exp\!\left(-\frac{1.293}{0.8}\right)\approx 0.20\approx\frac15 \]
The equilibrium ratio is frozen in at \(T_f\); this is the value the ratio would keep were neutrons stable. A
8
\[ \frac{n_n}{n_p}(t)=\left.\frac{n_n}{n_p}\right|_f\exp\!\left(-\frac{t}{\tau_n}\right),\qquad \left.\frac{n_n}{n_p}\right|_{\rm nuc}\approx 0.20\,e^{-260/880}\approx \frac17 \]
Between freeze-out and the onset of nucleosynthesis (\(t_{\rm nuc}\approx 3\)-\(4\) min, set by the deuterium bottleneck) free neutrons \(\beta\)-decay with lifetime \(\tau_n\approx 880\ \mathrm{s}\). B
9
\[ Y\equiv\frac{\rho({}^4\mathrm{He})}{\rho_{\rm baryon}}=\frac{4\,(n_n/2)}{n_n+n_p}=\frac{2\,n_n}{n_n+n_p}=\frac{2\,(n_n/n_p)}{1+(n_n/n_p)} \]
Each \({}^4\mathrm{He}\) nucleus consumes two neutrons and two protons; all neutrons are captured, and nucleon masses are nearly equal so number fractions become mass fractions. A
10
\[ Y=\frac{2\cdot\frac17}{1+\frac17}=\frac{2/7}{8/7}=\frac{2}{8}=0.25 \]
Substitute the post-decay ratio \(n_n/n_p\approx1/7\). A
\[ \boxed{\,Y=\frac{2\,(n_n/n_p)}{1+(n_n/n_p)}\approx 0.25\,}\qquad \frac{n_n}{n_p}\approx\frac17 \]

Reading. About a quarter of the baryonic mass of the universe emerges from the first minutes as \({}^4\mathrm{He}\) and three-quarters as hydrogen. The number is set almost entirely by the neutron fraction surviving to nucleosynthesis, which is itself pinned by the competition between the weak interaction and cosmic expansion near \(1\ \mathrm{MeV}\).

Units check. \(Y\) is a ratio of mass densities, hence dimensionless. Inside it, \(\Delta m c^2/k_BT\) is (energy)/(energy) and \(t/\tau_n\) is (time)/(time), both pure numbers, so the exponentials are well-defined and \(Y\in(0,1)\).

Limiting cases
  • \(k_BT\gg\Delta m c^2\) (very early): \(n_n/n_p\to1\), equal numbers of neutrons and protons.
  • Instant nucleosynthesis (\(t_{\rm nuc}\to0\), no decay): \(n_n/n_p\to1/5\) and \(Y\to 2/6=0.33\), the upper bound.
  • Faster expansion (larger \(g_*\)): higher \(T_f\), larger frozen ratio, larger \(Y\) — the extra-neutrino sensitivity.
  • \(\tau_n\to\infty\) (stable neutron): no decay correction, \(Y\to0.33\); \(\tau_n\to0\): all neutrons decay, \(Y\to0\).
  • \(\Delta m c^2\to0\): \(n_n/n_p\to1\) at all \(T\), driving \(Y\to0.5\) (equal He and H by mass).
Breaks when
  • Large lepton asymmetry. A degenerate electron-neutrino background (\(|\mu_{\nu_e}|\gtrsim k_BT\)) shifts the \(\beta\)-equilibrium so that \(n_n/n_p=e^{-\Delta m c^2/k_BT-\mu_{\nu_e}/k_BT}\); the simple Boltzmann initial condition and hence \(Y\approx0.25\) no longer hold.
  • Non-standard expansion at \(1\ \mathrm{MeV}\). Early matter domination, a fast-rolling scalar, or many extra relativistic species change \(H(T)\); freeze-out no longer occurs near \(0.8\ \mathrm{MeV}\) and the prediction fails.
  • Very low baryon density. If \(\eta\) were far smaller, the deuterium bottleneck would open too late, neutrons would have decayed away, and \(Y\) would collapse toward zero.
  • Sub-second inhomogeneity or new decays. Injecting energetic particles (decaying relics) after freeze-out can photodissociate \({}^4\mathrm{He}\) or reset \(n_n/n_p\), voiding the clean result.
Failure modes
  • Writing \(Y=n_n/(n_n+n_p)\) (a number fraction of neutrons) instead of the mass fraction of helium \(2n_n/(n_n+n_p)\) — off by a factor of two.
  • Using the freeze-out ratio \(1/5\) directly and forgetting the free-neutron decay to \(1/7\), which spuriously gives \(Y\approx0.33\).
  • Evaluating the Boltzmann factor with \(\Delta m c^2=1.293\ \mathrm{MeV}\) but \(k_BT\) in kelvin, or vice-versa — always keep both as energies.
  • Taking \(T_f\) as the temperature where nucleosynthesis happens; freeze-out (\(\sim0.8\ \mathrm{MeV}\)) and the deuterium bottleneck (\(\sim0.07\ \mathrm{MeV}\)) are widely separated in time and temperature.
  • Assuming \(Y\) depends strongly on \(\eta\); in fact \(Y\) is only logarithmically sensitive to the baryon density, unlike the deuterium abundance.
Discussion

The prediction rests on a race between two clocks. The weak-interaction clock \(\Gamma_{\rm weak}\sim G_F^2 T^5\) runs down steeply as the universe cools, while the expansion clock \(H\sim\sqrt{g_*}\,T^2/M_{\rm Pl}\) falls more gently. Their crossing near \(k_BT_f\approx0.8\ \mathrm{MeV}\) is where the neutron-to-proton ratio stops tracking equilibrium and is frozen at \(\approx1/5\). Everything downstream is bookkeeping: neutrons decay slowly, then get sequestered into the tightly bound \({}^4\mathrm{He}\) nucleus once deuterium can finally survive.

A deep feature is the near-cancellation of scales. The freeze-out condition \(G_F^2 T_f^5\sim T_f^2/M_{\rm Pl}\) gives \(k_BT_f\sim(G_F^2 M_{\rm Pl})^{-1/3}\), and it is a numerical coincidence of the Standard Model that this lands within a factor of order unity of \(\Delta m c^2=1.293\ \mathrm{MeV}\). Were \(T_f\) much larger than \(\Delta m c^2\), neutrons and protons would freeze out in equal numbers and \(Y\to0.5\); much smaller, and neutrons would be exponentially rare, \(Y\to0\). That \(Y\approx0.25\) reflects \(T_f\sim\Delta m c^2\), a soft anthropic-adjacent fine structure of primordial chemistry.

Helium is a cosmic clock and a cosmic scale. Because \(H\propto\sqrt{g_*}\), any extra relativistic species active at \(1\ \mathrm{MeV}\) raise the expansion rate, push \(T_f\) up, freeze in more neutrons, and increase \(Y\). This is why measured primordial helium constrains the effective number of neutrinos \(N_{\rm eff}\) and any light dark-sector particles; a single extra thermalized neutrino species shifts \(Y\) by roughly \(+0.013\), at the edge of observational reach.

The rigorous computation replaces the sudden-freeze-out approximation with a Boltzmann equation for \(X_n=n_n/(n_n+n_p)\), \(\dot X_n=-\lambda_{np}X_n+\lambda_{pn}(1-X_n)\), integrated through the smoothly varying rates with finite-temperature and Coulomb corrections, and couples it to the full nuclear reaction network. State-of-the-art codes yield \(Y_p=0.2470\pm0.0002\) for the CMB-inferred baryon density, in striking agreement with the analytic estimate. Common misconceptions: \(Y\) is not the fraction of atoms that are helium (that is \(\sim8\%\) by number) but the fraction of baryonic mass; and helium is not made in stars in this amount — stellar nucleosynthesis adds only a few percent on top of the primordial floor.

Worked examples
1
\[ \text{Freeze-out ratio: given } k_BT_f=0.80\ \mathrm{MeV},\ \Delta m c^2=1.293\ \mathrm{MeV}. \]
Evaluate the frozen neutron-to-proton ratio directly from the Boltzmann factor. A
2
\[ \frac{n_n}{n_p}\Big|_f=\exp\!\left(-\frac{1.293}{0.80}\right)=\exp(-1.616)=0.199 \]
Ratio of energies is dimensionless; \(e^{-1.616}\). A
3
\[ \text{Apply decay over } t_{\rm nuc}=260\ \mathrm{s},\ \tau_n=880\ \mathrm{s}:\quad \frac{n_n}{n_p}\Big|_{\rm nuc}=0.199\,e^{-260/880}=0.199\times0.744=0.148 \]
Free-neutron survival factor \(e^{-t/\tau_n}\). B
\[ Y=\frac{2(0.148)}{1+0.148}=\frac{0.296}{1.148}=0.258\approx 0.26 \]

Reading. The naive sudden-freeze-out estimate lands within a percent or two of the precise value \(0.247\); the small excess comes from the approximate \(T_f\) and \(t_{\rm nuc}\).

Units check. \(Y\) dimensionless, between 0 and 1.

1
\[ \text{Extra neutrino: how much does one added species raise } Y? \]
Standard model has \(g_*=10.75\) at \(1\ \mathrm{MeV}\) (photon, \(e^\pm\), three \(\nu\)); add one more neutrino flavor. B
2
\[ g_*\to g_*+\tfrac{7}{8}\cdot 2=10.75+1.75=12.5,\qquad \frac{T_f'}{T_f}=\left(\frac{g_*'}{g_*}\right)^{1/6}=\left(\frac{12.5}{10.75}\right)^{1/6}=1.025 \]
Since \(k_BT_f\propto g_*^{1/6}\) from step 6, a modest change. Neutrino contributes \(7/8\) per fermionic degree of freedom, \(\times2\) spin states. B
3
\[ k_BT_f'=0.80\times1.025=0.820\ \mathrm{MeV}\ \Rightarrow\ \frac{n_n}{n_p}\Big|_f'=e^{-1.293/0.820}=e^{-1.577}=0.207 \]
Recompute the frozen ratio at the higher \(T_f'\). A
4
\[ \frac{n_n}{n_p}\Big|_{\rm nuc}'=0.207\,e^{-260/880}=0.154,\qquad Y'=\frac{2(0.154)}{1.154}=0.267 \]
Same decay factor and helium formula. A
\[ \Delta Y = Y'-Y = 0.267-0.258 \approx +0.009\ \text{per extra neutrino species} \]

Reading. One extra thermalized neutrino species raises the helium yield by roughly one percent (full network gives \(\approx+0.013\)); this is the physical basis of the BBN bound on \(N_{\rm eff}\).

Units check. \(\Delta Y\) dimensionless. The \(g_*^{1/6}\) scaling makes \(Y\) reassuringly weakly dependent on the particle content.

Problems
  1. Compute the equilibrium neutron-to-proton ratio at \(k_BT=2.0\ \mathrm{MeV}\) and at \(k_BT=0.5\ \mathrm{MeV}\) (\(\Delta m c^2=1.293\ \mathrm{MeV}\)).
    SolutionAt \(2.0\ \mathrm{MeV}\): \(n_n/n_p=e^{-1.293/2.0}=e^{-0.647}=0.524\). At \(0.5\ \mathrm{MeV}\): \(n_n/n_p=e^{-1.293/0.5}=e^{-2.586}=0.0753\). The ratio falls from about \(1{:}2\) to about \(1{:}13\) as the universe cools through the freeze-out epoch — illustrating why the exact freeze-out temperature matters.
  2. Observations give \(Y=0.245\). Infer the neutron-to-proton ratio at nucleosynthesis.
    SolutionInvert \(Y=\dfrac{2r}{1+r}\Rightarrow r=\dfrac{Y}{2-Y}=\dfrac{0.245}{1.755}=0.1396\approx\dfrac{1}{7.16}\). So \(n_n/n_p\approx1/7\), consistent with freeze-out at \(\approx0.8\ \mathrm{MeV}\) followed by \(\sim4\) minutes of neutron decay.
  3. Using \(k_BT_f\propto g_*^{1/6}\), estimate the freeze-out temperature and \(Y\) if the universe had \(g_*=6.0\) instead of \(10.75\) at \(1\ \mathrm{MeV}\) (ignore the neutron-decay change; use \(t_{\rm nuc}=260\ \mathrm{s}\), \(\tau_n=880\ \mathrm{s}\)).
    Solution\(T_f'=0.80(6.0/10.75)^{1/6}=0.80\times0.912=0.730\ \mathrm{MeV}\). Frozen ratio \(e^{-1.293/0.730}=e^{-1.771}=0.170\). After decay: \(0.170\times e^{-0.295}=0.170\times0.744=0.127\). \(Y=2(0.127)/1.127=0.225\). A slower expansion (fewer species) lowers \(T_f\), freezes in fewer neutrons, and reduces \(Y\).
  4. If the neutron lifetime were \(\tau_n=300\ \mathrm{s}\) rather than \(880\ \mathrm{s}\), what helium mass fraction would result (take frozen ratio \(0.20\), \(t_{\rm nuc}=260\ \mathrm{s}\))?
    SolutionSurvival factor \(e^{-260/300}=e^{-0.867}=0.420\). Ratio at nucleosynthesis \(=0.20\times0.420=0.084\). \(Y=2(0.084)/1.084=0.155\). A shorter neutron lifetime lets more neutrons decay before capture, sharply lowering the helium yield — which is why \(\tau_n\) is a key BBN input and its precise measurement matters.
  5. The baryon-to-photon ratio is \(\eta\approx6\times10^{-10}\). Estimate the temperature at which deuterium stops being photodissociated, given \(B_D=2.22\ \mathrm{MeV}\), by requiring the number of photons above the binding energy per baryon to drop to order unity: \(\eta^{-1}e^{-B_D/k_BT_{\rm nuc}}\sim1\).
    SolutionTake logs: \(B_D/k_BT_{\rm nuc}\sim\ln(\eta^{-1})=\ln(1.67\times10^9)=21.2\). Thus \(k_BT_{\rm nuc}\sim B_D/21.2=2.22/21.2=0.105\ \mathrm{MeV}\); a more careful estimate using the photon spectral tail gives \(\approx0.07\ \mathrm{MeV}\). The enormous photon-to-baryon ratio is what delays nucleosynthesis far below \(B_D\), giving free neutrons the time to decay that sets \(n_n/n_p\approx1/7\).